System-level optimization and design of the high speed machining process using ceramic cutting tools

Materials and Design 21 Ž2000. 175᎐189

System-level optimization and design of the high speed machining process using ceramic cutting tools O. Sbaizero1, R. RajU Department of Mechanical Engineering, Uni¨ ersity of Colorado, Campus Box 427, Boulder, CO 80309-0427, USA Received 31 July 1999; accepted 14 September 1999

Abstract Research on ceramic cutting tools is interdisciplinary, straddling across engineering studies of stress analysis and heat transfer and materials science studies of microstructure-property relationships. A systems analysis methodology is used to investigate the linkages between these two disciplines. System-level partitioning of the functional relationships has identified three key linking variables: hardness, temperature and thermal diffusivity. The linking variables are used to estimate the flash temperature existing at the cutting interface. The flash temperature depends on the interaction between the change in the intrinsic hardness of the ceramic material with temperature and the influence of material hardness on the heat generated at the cutting interface. This information is used to construct process performance diagrams which may be useful in optimizing wear rate in the ceramic cutting tool against material removal rate and surface finish. The results are applied to two kinds of ceramic cutting tools: alumina and titanium carbide. The work piece is assumed to be AISI 4340 steel. The study highlights three key problems in materials science that need to be addressed in order to extend the microscopic understanding of materials to ceramic grinding and machining processes: Ži. micromechanistic understanding of temperature-dependence of hardness of ceramics; Žii. micromechanisms of wear and thermal shock behavior in the local, strongly varying temperature fields near the contact area between the tool and the work piece; and Žiii. microstructural understanding of thermal conductivity of cutting tool materials. 䊚 2000 Elsevier Science Ltd. All rights reserved. Keywords: Ceramic cutting tools; Machining processes

1. Introduction The optimization of engineering systems often depends on having a methodology that can integrate the basic understanding of materials behavior with engineering analysis. A system level approach for developing such a methodology was presented in a recent paper w1x. This approach begins with partitioning the whole problem into subsystems and linking variables. The linking variables serve as the means for studying the interaction between the subsystems. The partition-

1

On leave from the University of Trieste, Italy. Corresponding author. Tel.: q1-303-492-1029; fax: q1-303-4923498. E-mail address: [email protected] ŽR. Raj.. U

ing of the system into subsystems is carried out with a graphical optimization routine w2x. Interestingly, the subsystems identified by this partitioning process are often found to be consistent with traditional disciplines, such as mechanical analysis, engineering analysis and materials science. Thus, the methodology builds on the existing foundations of different disciplines, seeking linkages that can lead to new interdisciplinary directions and innovations. The overall objective of this approach is to enhance performance and cost effectiveness of systems by incorporating latest research and developments in materials science. The specific objectives of the integrated analysis are probabilistic life prediction w3x, optimization and design w4x. The present paper addresses the problem of high

0261-3069r00r$ - see front matter 䊚 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 1 - 3 0 6 9 Ž 9 9 . 0 0 0 6 7 - 9

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speed machining with ceramic cutting tools. We address the issue of optimization and design of the machining process in order to enhance performance.

2. The system-level approach

The system integration approach being used in this and the earlier papers w1,3x consists of three steps: 1. Reviewing the current status of the overall problem both from the engineering analysis as well as the materials science perspective. The results from models, and from empirical measurements are collected. The variables and the equations are grouped into independent and dependent variables, as summarized in Fig. 1. The independent variables are often related to materials selection, microstructure design, and the geometrical configuration of the system. The dependent variables usually relate performance of a system to the independent variables. The dependent variables include analytical themes such as micromechanical modeling of material behavior, stress analysis, heat transfer analysis and structural engineering.

2. In the second step the functions and variables are partitioned by a graphical method w2x into subsystems and linking variables. Partitioning may be extended to several levels until a subset of equations that can be used as the basis for the performance analysis, emerge. This process of downselection of functions and variables is illustrated by the flow chart in Fig. 2. 3. In the final step the subsystems are analyzed to achieve the specific results at the system-level. These objectives may include life prediction, optimization and design. The present paper is focussed on optimization of the process parameters, for example, cutting speed, material removal rate, surface finish, in order to achieve the lowest possible wear rate in the ceramic cutting tool. Each of the three steps summarized just above are applied to the high-speed machining problem in the following sections. Two kinds of ceramic cutting tools, made from either polycrystalline alumina ŽPCA. or titanium carbide ŽTiC. are considered. The work piece material is assumed to be AISI 4340 steel, in both cases. The system-level partitioning process reveals gaps in our knowledge for mechanistic linkage between materials science studies and the engineering practice of the

Fig. 1. The procedure for assembling and grouping of functions and variables.

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Fig. 2. Flow chart showing the process of partitioning into subsystems at many levels, and using this information for down selection of functions and variables for system-level analysis.

high speed machining process. These critical issues are discussed towards the end of the paper.

3. Materials science and engineering analysis studies The publications relating to high speed machining process with ceramic cutting tools, may be broadly classified into three areas: 1. The first area includes work related to the basic microstructure property relationships of ceramic materials that are perceived to be relevant to the application of ceramics in high speed machining. Examples of such studies include the estimate of thermal shock resistance w4,5x and wear resistance w6,7x in terms of the fracture toughness, the hardness, thermal conductivity and thermal expansion coefficient and the microstructure of the ceramic. Other studies report the temperature and grain size-dependence of hardness w8x and estimates of thermal expansion coefficient w9x, the thermal conductivity w10x and elastic moduli w11x of two phase microstructures. 2. Results of engineering analysis that study the forces and energetics of the machining process are included in the second group. Examples of papers in this area include the estimate of the force exerted on the cutting tool as a function of the feed rate, depth of cut and cutting velocity w12,13x, and the calculation of the interface temperature, also called

the flash temperature, attained during the cutting process w14,15x. 3. A third set of studies aims to study the overall performance of the process, and includes the following types of analyses: tool life w16x; wear rate of the ceramic tool w17x; surface finish w18x; and productivity index w19x. As pointed out by the flow chart in Fig. 1, the variables in the problem may be classified into independent variables and dependent variables. Independent variables generally define the ground level parameters in the problem, for example, the choice of materials Žas well as their microstructure . for the cutting tool and the workpiece are considered to be independent variables. Also, we shall assume that the geometrical configuration of the tool and the workpiece are held constant; therefore, parameters related to geometry are treated as independent variables. Following the earlier nomenclature w1x, variables describing the fundamental material properties are designated by m, the microstructure by s, and the geometrical configuration of the system by z. The dependent variables are related to the independent variables via functions. In general these variables may be subdivided into three categories: materials science Ž r ., engineering analysis Ž y ., and performance Ž q .. The materials science variables describe the engineering properties of materials in terms of their fundamental properties; for example the temperature-dependent hardness of the material may depend on its

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microstructure and upon its fundamental properties such as elastic moduli and diffusion coefficients. In engineering analysis we seek to identify a process condition in terms of the basic processing parameters; for example, the force exerted on the tool as a function of the cutting velocity and material removal rate. The system functions describe the relationships between independent and dependent variables. Therefore, the classification of the functions follows the classification of the dependent variables, into relationships that describe material properties Ž R ., those that describe the engineering behavior of the process Ž Y ., and those that articulate the performance characteristics of the process Ž G ..2 Table 1 lists the variables, and Table 2 gives the functional relationships that have been gathered from the studies that we discussed in Section 2. In both tables the original nomenclature for the variables is included. The functions have been numbered sequentially so that they can be referred to more easily in this paper. In the next section we move to the second step of the system-level analysis, that is, partitioning of the functions and variables into first level, and second level subsystems that are related to one another by linking variables. We shall find that the linking variables not only provide the mechanism for building bridges between the subsystems, but also provide physical insights into the relationship between materials science issues and engineering performance.

4. Partitioning of functions and variables The purpose of partitioning is to group the variables and functions in such a way that the connectivity between equations is revealed. In non-hierarchical partitioning w1x, which is of interest here, the subsystems are linked to one another through linking variables. Generally, the objective of partitioning algorithms is to reduce the number of linking variables to a minimum. A first level partitioning may be further partitioned into a second tier of subsystems in order to gain further insights for analysis and optimization. The problem considered in this paper will employ such multi-level partitioning. The results of the partitioning analysis are summarized in Fig. 3. Note that these results are not unique ᎏ there may be other ways of partitioning the overall

2 In reference w1x the symbol ⌫ was used to describe the G type functions.

Table 1 Independent ¨ ariables Fundamental material properties ¨ ariables (m) ␣m Matrix thermal expansion coefficient ␣r Reinforcement thermal expansion coefficient km Matrix thermal conductivity Em Matrix Young’s modulus Er Reinforcement Young’s modulus ¨r Reinforcement Poisson’s ratio ¨m Matrix Poisson’s ratio H0,m Matrix room temperature hardness H0,r Reinforcement room temperature hardness Tm Melting temperature KIcm Matrix toughness ␴y ,r Reinforcement yield strength a1 Constant in matrix hardness equation a2 Constant in reinforcement hardness equation b Constant in tool life equation Y1 Constant in flank wear equation. Y2 Constant in crater wear equation ⌫ Constant in cutting tool life equation Z Constant in wear rate equation x1 Constant in flank wear equation x2 Constant in crater wear equation ¨1 Constant in flank wear equation ¨2 Constant in crater wear equation A Constant in wear rate equation B Constant in wear rate equation n Constant in wear rate equation s Constant in composite Young modulus eq. Q Constant in composite Young modulus eq. ␩ Constant in matrix hardness equation ␭ Constant in reinforcement hardness equation dc Critical grain size for spontaneous microcracking rc Kapitza radius ␹ Constant in wear rate due to pullout eq. c Composite-specific heat ␳ Composite density M Molar volume Hw p Workpiece hardness ⌬G Free energy of formation uU Reinforcement strain to failure Microstructure properties ¨ ariables (s) f Reinforcement volume fraction D Reinforcement grain size d Matrix grain size Geometrical configuration of the system ¨ ariables (z) rt Tool nose radius ␮ Friction coefficient ␥ Cutting angle Dependent ¨ ariables Material science ¨ ariables (r) ␣c Composite thermal expansion coefficient Ec Composite Young’s modulus Hr Reinforcement hardness Hm Matrix hardness ␦ Nearest neighbor distance ␤ Kapitza radius and reinforcement grain size ratio KIcc Composite toughness RH Composite thermal shock resistance

O. Sbaizero, R. Raj r Materials and Design 21 (2000) 175᎐189 Table 1 Ž Continued. RW ␴ mR kc HC C ¨c

Composite abrasive wear resistance Residual stress in the matrix Composite thermal conductivity Composite hardness Chemical solubility Composite Poisson’s ratio

Engineering analysis ¨ ariables (y) ⌬T Critical temperature difference ␴ mT Tensile stress in the matrix due to sliding contact ␴ mD Tensile stress in the matrix induced by accumulated damage Tf Cutting tool flash temperature T0 Average cutting tool temperature Fy Thrust force Žradial. on the tool VB Flank wear KT Crater wear h Chip thickness Performance ¨ ariables (q) DOC Depth of cut F Feed rate V Cutting velocity t Cutting time Wp,0 Wear volume due to pull-out WR The total wear rate tU Tool replacement time ⍀ Tool life R Surface finish ⌰ Material removal rate P Productivity

system. Fig. 3 shows two main subsystems: materials science ŽM. and engineering analysis and performance ŽE., which are linked by three variables, the thermal conductivity, the hardness, and the temperature. This result illustrates the importance of thermal conductivity of the ceramic material in the machining process. In the present article we consider only simple ceramic materials, PCA and TiC, and therefore the thermal conductivity will be held constant, and not considered further in the optimization analysis. The partitioning process identified a third subsystem, shown at the bottom right in Fig. 3. It consists of functions that do not share any variables with the other functions in the system. It does not mean these functions are not pertinent to the system. It means that these functions are ‘orphans’, and a deliberate effort is required to define their role in the overall systems analysis. The subsystems M and E are further partitioned into M1᎐M3, and E1᎐E4. The E subsystems share two linking variables, the cutting velocity and the chip thickness. The existence of second level subsystems implies that the performance analysis may be simplified by choosing between these subsystems. This choice is guided by measuring the connectivity factors for the functions to the linking variables.

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The connectivity factor measures the direct and indirect relationship of a function to the linking variables. Type A connectivity is obtained when the linking variable is included in the function, and Type B when the function is related to a linking variable via another function. The connectivity factors for the functions are shown in the two right hand columns in Fig. 3. For example R7 linked directly to hardness giving a Type A connectivity factor of one. Similarly, Y 1 has a Type A connectivity factor of two since it is linked to both temperature as well as hardness. As an example of Type B connectivity, R1 is connected to hardness via the function R7; since equations R1 and R7 share two variables, K c and ␦ , we obtain a Type B connectivity factor of two. Tertiary level connectivities to the linking variables can be considered if the first and second order connectivity results do not provide enough distinction between the relative importance of subsystems. From this information we choose those functions which have Type B connectivity factor of, at least, one or Type II connectivity factor of, at least, four. This down selection process leads to the matrix given in Fig. 4 with two main subsystems. The subsystems in Fig. 4 are partitioned further. One may choose between these subsystems for the performance analysis. Here, the objective is to include as many functions as possible, therefore, we choose the subsystems that are shown in Fig. 5 for final analysis. Fig. 5 gives the equations that are used in the performance analysis. The equations from the materials science subsystem, R8, R9 and R10 describe the temperature-dependence of the hardness of a two-phase ceramic material. Since we are considering only singlephase materials, PCA and TiC, we need to know how the hardness of these materials varies with temperature. The equations from the engineering analysis subsystem, Y 1 and Y 2 describe the estimate of the flash temperature at the cutting tool-work piece interface in terms of the process variables. The equations from the performance subsystem, R11 and G3, describe the chemical and abrasion wear components of the tool life in terms of temperature and the process variables. In Section 5 we develop performance diagrams based upon these sets of equations.

5. Performance diagrams Performance diagrams for two materials systems are developed: PCAr4340 and TiCr4340, that is the cutting tool is made from polycrystalline alumina or from titanium carbide, whereas the work piece is an iron alloy containing 3% carbon ŽAISI 4340.. This section is divided into three parts. The first describes the procedure used to normalize the variables; this is necessary since the equations are written

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Table 2 Functional relationships used in the partitioning Function

KIcc s KIcm q

8␲ uU␴y,r ␦ 1q D

KIcc Ž 1 y ¨ c . kc Ec ␣c

RH s

1

kc s km Ec s

)

KIcm q

Ž1 y f .

Ž . 3 1y ␤ r

Em Er 1 wŽ1 y f . Em q fEr x4 q 2 f Em q Ž 1 y f . Er

ž

/

½

w 1 y Q Ž TfrTm . s x 1 y QTms

5

y1

Em ␣m Ž 1 y f . q Er ␣r f Em Ž 1 y f . q Er f

RW s

Description

Ref

R1

Composite toughness

w20x

R2

Composite thermal shock resistance

w4,5x

R3

Composite thermal conductivity

w10x

R4

Composite Young’s modulus

w11x

R5

Residual stress in the matrix due to sintering

w21x

R6

Composite thermal expansion coefficient

w9x

R7

Composite’s abrasive wear resistance Matrix hardness as a function of temp. and grain size Reinforcement hardness as a function of temp. and grain size Composite hardness Equilibrium solubility of tool material in the workpiece

w6,7x

Ž 1q 2 ␤ .

1 q 2¨m 1 y 2¨r 1 ␴mR s Ž ␣m y ␣r . ⌬T q 2 2 Em Er

␣c s

Equation number

KIcc1r 2 Hc5r8 1 Ž EcrHc . 4r5 ␦

Hm s H0,m q ␩ dy1 r2 expya iT f

R8

Hr s H0,r q ␭ Dy1 r2 expya 2 T f

R9

Hc s fHr q Ž1 y f . Hm C s Aey⌬GrRT

R10 R11

w8x w22x w23x w8x

␦s

2 1yf D 3 f

R12

w6x

␤s

rc D

Nearest neighbor distance between the reinforcing particles

R13

Ratio between Kapitza radius and reinforcement grain size

w10x

R14

Tensile stress in the matrix induced by accumulated damage Flash temperature on the cutting tool Thrust force Žradial. on the cutting tool

w24x

G1

Surface finish

w18x

G2 G3

w16x w17x

G4 G5 G6

Cutting tool life Abrasive and chemical dissolution wear rate Width of the flank wear land Width of the crater wear Material removal rate

G7

Productivity

w19x

G8

Chip thickness

w19x

G9

Wear rate due to pull out

w24x

ž /

␴mD s ␴m R

d dc

ž /

1r2

y1

Tf s T0 q 0.3 ␮ V 1r 2 Fy0.25 Ž␲ Hc .3r4 Žkc ␳ c.y1r2

Y1

Fy s 1023.24V y0 .183 F 0.393 DOC 0.465

Y2

Rs

F2

18'3 rt ⍀ s ⌫Tfyb WR s AVZŽ Hw prHc .ny1 Ž1rHc . q BMCV 0.5 VBs Y1t x iVi KTs Y2 t x 2V2 MR s 12 F DOC V hV⍀ Ps ⍀ y tU DOC F sin ␥ hs DOCy rt Ž 1 y cos␥ . q rt ␥ q F sin Ž ␥r2. Wp .o.s ␹

d dc

ž /

␴mT q ␴mD q ␴mR ␴mR

y

d dc

ž /

1r2

in complex units. The second section discusses the overlay of the equations from the materials science and the engineering analysis subsystems ᎏ the main result here is that the actual flash temperature at the machining interface is determined by the intersection between the intrinsic temperature-dependence of the hardness

w14x w15x w12x w13x

w25x w25x w26x

of the ceramic material and the influence of hardness on the heat generated at the machining interface. In the third section the dynamic values of the hardness and temperature at the machining interface are used to calculate the performance diagrams which show the relationships between the cutting velocity, material re-

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Fig. 3. First and second level partitioning of the functions.

moval rate and chemical and abrasion wear rate of the cutting tool. We include the surface finish parameter in these diagrams as well, note that the significance of this parameter is revealed not by partitioning analysis but rather by heuristic reasoning. 5.1. Normalization of the ¨ ariables The equations needed for the performance analysis involve complex units. Therefore, we normalize the variables with respect to standard values. The process variables in the system are the cutting tool velocity, V, the feed rate, F, the depth of cut, DOC, and the material removal rate, MR . The engineering analysis parameters are the flash temperature, Tf , and the thrust force on the tool, Fy ; we shall find that Y 1 and Y 2 can be combined to eliminate Fy as a running variable. The materials science variable is hardness of the cutting

tool and of the workpiece. The performance variables are the chemical wear rate and the abrasion wear rate of the ceramic tool, and the surface finish parameter. The normalizing values used for these variables are given in Table 3. The normalizing values are distinguished by a superscript, ‘o’, except for temperature where TM , the melting temperature of the ceramic, is used for normalization. The normalized values are defined by enclosing the variables between angular brackets, that is, ² V :, ²T :, ² H :, etc. for example: ² H :s

H Actual hardness s , Hardness at room temperature Ho

² Hf : s

Hf Hardness at the flash temperature s , Hardness at room temperature Ho

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Fig. 4. Down selection of functions from second level partitioning.

²T : s

²Tf : s

T , Temperature in K , TM , Melting temp of the ceramic material

5.2. O¨ erlay of materials science and engineering analysis

Flash temperature in K , TM , Melting temp of the ceramic material

In the initial step we consider the overlay of results from the materials science subsystem, R8, R9 and R10, with the engineering analysis subsystem, Y 1 and Y 2. The principal materials science input is the intrinsic temperature-dependence of the hardness of materials.

Ž1. etc.

Fig. 5. The final partitioning used in performance analysis.

O. Sbaizero, R. Raj r Materials and Design 21 (2000) 175᎐189 Table 3 Normalizing values used for the chosen variables needed for the performance analysis

Cutting tool velocity, V⬚ Žmrmin. Feed rate, F⬚, Žmmrrev. Depth of cut, DOC⬚ Žmm. Average cutting tool temperature, T⬚ ŽK. Melting temperature, Tm ŽK. Cutting tool wear rate, W⬚ Ž ␮ mrmin. Hardness at room temperature ŽGPa. a1 constant in Eq. Ž6. a2 constant in Eq. Ž8. n constant in Eq. Ž8. w17x

PCAr AISI 4340

TiCr AISI 4340

500 0.32 2 1023 2323 100 15.3 18.6 2.594 7

183 0.32 2 1418 3413 0.87 32 y2.90 0.511 7

As far as we know fundamental mechanistic models that describe the temperature-dependence in terms of fundamental microstructural parameters and thermally activated processes, such as diffusion, are not available for the materials systems being considered in this paper. Therefore, we resort to empirical relationships. Using the normalization’s given in Table 3 we obtain

183

the following equations for intrinsic temperature-dependence of polycrystalline alumina, titanium carbide, and AISI 4340 iron alloy w27,28x: PCAw27x :² H : s 15.5exp Ž y0.0012²T :. TiC w28x :² H : s 3.3exp Ž y0.00183²T :. from 273 K to 1273 K TiC w28x :² H : s 27exp Ž y0.00393²T :. from 1273 K to 1773 K AISI w28x 4340:² H : s 12exp Ž y0.00163²T :.

Ž2.

Plots of Eq. Ž2. are given in Fig. 6. The change in the hardness equation for TiC at approximately 1273 K is due to a phase change. At low temperatures the behavior is characteristic of covalent bonding, whereas at higher temperatures there is a change from predominantly covalent to metallic bonding in the material w29x. The equations from engineering analysis are com-

Fig. 6. Empirical information showing the change in the intrinsic hardness of cutting tool and workpiece materials.

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bined by substituting for Fy in Y 2 into Y 1. Furthermore, we neglect To , the ambient temperature, in comparison to the flash temperature. Taking logarithms of both sides of the equation and using the normalizations in Table 3, we obtain the following result: ln²T : s A⬘ y 0.5ln² ␬ : q 0.45ln² V : q 0.1ln² F : q 0.12ln² DOC : q 0.75ln² H :

Ž3.

Here A⬘ contains the constants in Y 1 and Y 2, and ␬ is the thermal diffusivity, so that Ž ␬ s k⬘c ␳ c ., where k⬘c , is the thermal conductivity, ␳ is the density and c is the specific heat at constant pressure. Eq. Ž3. is further modified by using the equation for the material removal rate, G6: ln² MR : s A⬙ q ln² V : q ln² F : q ln² DOC :

Ž4.

where A⬙ is a constant of normalization. Combining Eq. Ž3. and Eq. Ž4. and making the approximation that 0.02 ln² DOC : < 1 Žthis is reasonable since in most operations the depth of cut is kept nearly constant., we obtain: ln²T : s A q 0.35ln² V : q 0.1ln² MR : q 0.75ln² H :

Ž5.

where A includes not only the constants A⬘ and A⬙ but also the term ln² ␬ : since the thermal diffusivity of the materials is not expected to change significantly with temperature. It now remains to assign a value to A. We do this by comparing Eq. Ž5. to experiment where the flash temperature has been measured for the conditions defined by the normalizing parameters in Table 3. This comparison leads to a value of A s y0.163 for PCArAISI 4340 w30x and A s 0.897 for TiCrAISI 4340 w17x. The overlays of the materials science information in Fig. 4 with the prediction from Eq. Ž5. are given in Fig. 7a,b. Logarithmic scales are used for ² H : and ²T : so that Eq. Ž5. can be expressed as straight lines. The curves in Fig. 6 have been redrawn using normalized variables. The intersection between the results from the materials science and engineering analysis subsystems gives the solution for the flash temperature. For example for ln² MR : s 0.5 and ln² V : s 0.4 in the PCArAISI 4340 system, the flash temperature is predicted to be ln²Tf : s y0.7216 or 1129 K, and H f s 5.6 GPa. The procedure shown in Fig. 7 can be used to obtain the flash temperature for a range of process parameters. This information is now used to construct the performance diagrams as described in the next section.

Fig. 7. Cross-over between materials science and engineering analysis subsystems to determine the flash temperature and the toolworkpiece interface for different sets of materials removal rate and cutting velocity conditions Ža. Tool, PCA, Workpiece, AISI 4340. Žb. Tool, TiC; Workpiece, AISI 4340.

5.3. Construction of the performance diagrams The coupling between the engineering analysis and materials science leads to the estimate of the flash temperature at the machining interface. The local hardness of the ceramic cutting tool material and the work piece, at this temperature, is obtained by Eq. Ž2.. This information is substituted in Eqns R11 and G3 to calculate the wear rate of the cutting tool. Later, we shall overlay the surface finish parameter, given by G1, to the performance diagrams. We employ normalized variables and logarithmic forms of the functions in order to deal with the different kinds of units that are used in the high speed machining process and in materials science. The use of logarithmic functions requires us to separate the two

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terms in G3, the first representing abrasion wear and the other chemical wear, which we write as Wab and Wch , respectively. Substituting from R11 into Wch , we obtain the following form for ²Wch :: ln²Wch : s a1 q 0.5ln² V : y

⌬G 1 ⭈ RTM ²Tf :

Ž6.

where ⌬G is the free energy for the reaction between the cutting tool material and the workpiece, and R is the gas constant. TM is the melting temperature of the cutting tool. For a given workpiece material and cutting tool of composition A x B y , the term ⌬G can be calculated as follows w29,31x. ⌬Gs

⌬GA x B y q x⌬GA q y⌬GB xq y

Ž7.

where ⌬GA x B y is the free energy of formation of the cutting tool, ⌬GA is the relative partial molar excess free energy of solution of component A of the tool into the workpiece material, ⌬GB is the relative partial molar excess free energy of solution of component B of the tool into the workpiece material w31x. The equation for the abrasive wear rate, derived from the first term in G3 is as following: ln²Wab : s a2 q ln² V : q Ž n y 1 . ln²

Hw p : y ln² H f : Hf Ž8.

The quantities a1 and a2 in Eqs. Ž6. and Ž8. are constants of normalization. Two pieces of experimental data are used to determine these constants. First, the process variables where abrasion wear is found to be equal to chemical wear Žor a known fraction thereof. is used to calculate the quantity Ž a1 y a2 ., obtained by setting Eq. Ž6. equal to Eq. Ž8.. The second piece of information is the measurement of the wear rate under a different set of conditions, which gives either a1 or a2 depending on which mechanism is dominant. The values of a1 and a2 so obtained for the two material systems are given in Table 3. For PCA a1 was calculated based on the assumption that at 1023 K the chemical wear is only f 5% of the total wear w30x. In the case of TiC, a1 was calculated assuming that at 1140 K the contributions of abrasion and chemical wear to the cutting tool wear are equal w17x. The rates for the chemical and abrasion wear for the two materials systems are compared in Fig. 8. Interestingly, chemical wear is greater than abrasion wear at temperatures below 1150᎐1070 K in TiCrAISI 4340. In the case of PCAr4340 chemical wear dominates at

Fig. 8. Rates for the chemical and abrasion wear, for different material removal rates and cutting velocities Ža. tool, PCA, workpiece, AISI 4340. Žb. tool, TiC, workpiece, AISI 4340.

higher temperatures, that is above 1220᎐1290 K. The reason for this difference is that the ratio of the hardness of the work piece and the ceramic changes much more rapidly with temperature for TiCr4340 than for PCAr4340. Another reason is that heat of chemical solubility of the workpiece in the ceramic material is smaller for TiCr4340 than for PCAr4340. Generally, the high speed machining process should be carried out in a regime where abrasive rather than chemical wear is the dominant mechanism. Therefore, it is predicted that the operating regime for TiCr4340 lies above 1125 K, while the operating regime for the PCAr4340 system lies below 1325 K. The performance diagrams developed below reflect these regimes of operation. The performance diagrams are constructed with the

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aid of equations that describe the abrasion wear of the ceramic tool wEqs. Ž2., Ž5. and Ž8.x. We also use G1 to describe the surface finish; rewriting G1 in terms of ² MR : and ² V :, by substituting from G6 and making the assumption that DOC is maintained nearly constant, we obtain: ln² R : s bq 2ln² MR : y 2ln² V :

Ž9.

where b is an arbitrary constant; we set it equal to unity by assuming that ln² R : s 1 when ² MR : s 1 and ² V : s 1. Note that a smaller value of R implies a better surface finish. 5.4. Results The performance diagrams are constructed to show the change in the wear rate with the materials removal rate, MR , the cutting velocity, V, and the surface finish, R. Since temperature is such an important consideration in the analysis, we include it in the diagrams as well. Of course, other ways of plotting the information may be more useful in certain situations; these can be constructed for specific applications, but for the present, we believe that the diagrams given here provide useful engineering guidelines as well as physical insights into the high speed machining process. The graphs for PCAr4340 are shown in Fig. 9a. They contain the lines of constant velocity, V, constant material removal rate, MR and constant surface finish, R. The wear rate and flash temperature serve as the axes. Lines for constant V are shown by dashed lines. The solid lines represent R, and the dotted lines show constant values of MR . The shaded areas in Fig. 9a represent different regimes of process parameters. Each shaded region represents a minimum value for the material removal rate and, at least, a certain level of surface finish, which is defined by a maximum value for R. Each regime also constrains the cutting velocity to lie above a critical value, since this is a feature of the high speed machining process. All three criteria are satisfied in each of the shaded regions, with every region corresponding to a different set of constraints on MR , R, and V. An unusual feature of the results in Fig. 9a is that a higher material removal rate produces a lower wear rate in the cutting tool. The reason for this result lies in the form of Eq. Ž8.. As expected, the flash temperature increases with increasing material removal rate; but the equation for abrasive wear depends on the hardness ratio of the workpiece and the cutting tool. Since the value of ‘n’ in Eq. Ž8. is assumed to be 7, the term containing the hardness ratio usually dominates over the last term, which depends only on the hardness of the cutting tool. This last term always increases abrasion wear with an increase in the flash temperature

Fig. 9. Ža. Performance diagram for high speed machining of AISI 4340 steel with ceramic cutting tools made from polycrystalline alumina. Each of the shaded areas shown above represent a regime where the material removal rate Ž MR . is greater than a certain value and the surface finish exceeds a given level, and where the cutting velocity lies above a critical value. The depth of cut is assumed to be approximately 2 mm. Žb. Comparison between the experimental data points and results predicted by Eqs. Ž5. and Ž8. for PCArAISI 4340. The experimental data derives from references w30,32x and w33x.

because of the negative sign, but the third term which depends on the hardness ratio may increase or decrease with temperature. In the case of PCArAISI 4340 the term containing the hardness ratio dominates leading to a lower wear rate at higher temperatures. Certainly, the validity of Eq. Ž8. can be questioned since it is based on empiricism; still if it is correct the performance diagrams suggest using a very high material removal rate for better surface finish and lower cutting tool wear. More importantly, the work presented here emphasizes the need for mechanistically based equations for abrasive wear that can be relied

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upon to provide accurate prediction for use in the performance diagrams. The equations used to construct Fig. 9a contain only two adjustable parameters, a1 and a2 , which were determined by two conditions: Ža. the condition where mechanical and chemical abrasion rates are equal; and Žb. an experimental data point in the abrasion dominant region. The predictions from the equations with these values for a1 and a2 are compared with experimental data in Fig. 9b. The point at V s 500 mrmin and MR s 800 cm3rmin was used as condition Žb. and, therefore, shows exact overlap of prediction and experi-

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ment. All other points compare the predicted extrapolations with experimental measurements of wear, except for one point at very low velocity Ž165 mrmin. the agreement is very good. The discrepancy at this low velocity may be attributed to a lack of adiabatic conditions that are assumed to exist in high speed machining. Returning to Fig. 9a, the best performance for the cutting tool, that is the minimum wear rate, is obtained at the lowest point of the shaded triangle. For example in the case of the shaded triangle on the right, the best performance is obtained at the point marked by a star.

Fig. 10. Ža. Performance diagram for high speed machining of AISI 4340 steel with ceramic cutting tools made from titanium carbide. Each of the shaded areas shown above represent a regime where the material removal rate Ž MR . is greater than a certain value and the surface finish exceeds a given level, and where the cutting velocity lies above a critical value. The depth of cut is assumed to be approximately 2 mm. Žb. Comparison between the experimental data points and results predicted by Eqs. Ž5. and Ž8. for TiCrAISI 4340. The experimental data derives from references w17x and w34x.

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This point prescribes that MR ) 500 cm3rmin, and cutting velocity of V ) 112 mrmin, would lead to a surface finish of R - 0.000004 and a wear rate of WR s 33.3 ␮rmin. Under these conditions flash temperature is expected to be 658 K. The depth of cut for all conditions is assumed to be 2 mm; a change in depth of cut by a factor of 2 will produce only approximately a 10% change in the ln²W : value for the wear rate of the cutting tool. However, the depth of cut is likely to influence the surface finish, but this effect still needs to be expressed in a quantitative form. The corresponding graphs for TiCr4340 are given in Fig. 10a,b. The windows for optimum operation have a different shape than in Fig. 9 because the abrasion rate changes very rapidly with temperature. In this case the wear rate does increase with higher material removal rate; the underlying reason being that the term containing the hardness ratio in Eq. Ž8. increases at higher temperatures, leading to a higher rate of abrasion. The shaded regions that define an operating regime in Fig. 10a have a different shape to those in Fig. 9a. This difference results from the relative magnitudes of chemical and abrasion wear for the two systems. In PCAr4340 the abrasion wear dominates at the lower temperatures, while in TiCr4340 abrasion wear dominates in the high temperature regime. The predicted wear rate is compared to the experimental data in Fig. 10b. The point at MR s 1405 cm3rmin and V s 183 mrmin was used to fix one of the two adjustable parameters. The agreement between extrapolation and experiment is excellent over a widerange of cutting velocity and material removal rates.

6. Critical materials science issues In this section we ask the following question: why have the microstructural parameters of the cutting tool materials not played a role in the quantitative analysis presented here? For example, only equations R8, R9 and R10 from the materials science subsystem were used in the construction of the performance diagrams. The answer emerges by inspecting the partitioned matrix in Fig. 3: the performance functions, G, are linked to materials science only through hardness and temperature. It is interesting to note that while more sophisticated micromechanisms for ceramic wear have been studied Žfor example R7. the understanding of how the intrinsic hardness of materials depends on temperature and microstructure remains a neglected area of fundamental research. The partitioned matrix in Fig. 3 provides useful guidelines to develop an intimate link between fundamental materials science research and its application to tool wear and machining efficiency. In general, we show that this linkage requires a study of the local wear

behavior, near the contact interface where the temperature gradients are very large. Studies of wear that invoke thermal shock, fracture toughness, and thermal diffusivity in the material on the local scale of the hot-spot that exists at the machining interface, are needed. These studies can build on the fundamental work that is already in place Žfor example R1, R2 and R7. but they must be extended to the spatially distributed temperature that is an essential feature of the high speed machining process. Investigation of issues such as comparing the length scale of the grain size with the length scale of the temperature distribution can be intriguing and rewarding, giving new insights in the development of ceramic materials for cutting tool applications. Finally, this study identifies the understanding and characterization of thermally diffusivity of ceramic materials as a key area of scientific study. Microstructural studies of thermal conductivity remains an area where new work can have a significant impact in the fields of high speed machining and ceramic grinding.

7. Conclusions

1. The partitioning of functions and variables from materials science and engineering studies of high speed machining has identified the intrinsic temperature dependence of hardness as the key, linking variables between these two fields of study. 2. The flash temperature that exists at the machining interface is determined by the crossover between the intrinsic hardness of the material and the influence of hardness on heat generation at the machining interface. The dynamic condition is determined by this balance between hardness and temperature, both from a materials science and an engineering analysis point of view. 3. Equations were developed to calculate the above ‘flash temperature’ for different conditions of cutting velocity and material removal rate for two cases, one using polycrystalline alumina and the other using titanium carbide as the cutting tool materials. The work piece in both instances was assumed to be AISI 4340 steel. 4. The above information was used to estimate the wear rate of the cutting tool, and to predict surface finish on the work piece. The wear rate was separated into two parts: chemical reaction and abrasion. In the normal operating range it was determined that abrasion wear dominated at higher temperatures for titanium carbide but at lower temperatures for polycrystalline alumina. 5. Performance diagrams have been constructed that delineate the wear rate and the surface finish as a

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function of the two most critical process variables: cutting velocity and material removal rate. The diagrams can prescribe the process condition to achieve a material removal rate and surface finish at an acceptable level of tool wear rate. The diagrams show that considerable reduction in wear rate can be achieved by optimum choice of the cutting velocity. 6. The partitioning results show that materials science studies that consider wear, thermal shock and deformation under the strong thermal gradients existing at the contact area between the tool and the workpiece are needed to extend the current understanding of material behavior to the high speed machining and the ceramic grinding processes. 7. The partitioning results also show that the study of microstructural origins of thermal conductivity in ceramic materials would be needed for significant science-based advances in this technology.

Acknowledgements This work was supported by the Division of Materials Research at the National Science Foundation under Grant No: DMR-9796100, and by the Department of Mechanical Engineering, the College of Engineering and the Graduate School of the University of Colorado at Boulder. It is a pleasure to thank Luca Nardone for his help in analysis and construction of the performance diagrams. OS acknowledges financial support from the Italian National Research Council ŽCNR. under the grant PF MSTA II. References w1x Subbarayan G, Raj R. A methodology for integrating materials science with system engineering. Mater Des 1999;20:1᎐12. w2x Henderson B, Leland R. The Chaco user’s guide, version 2.0 ᎏ technical report SAND85-2344, Sandia National Laboratories, Albuquerque NM. w3x Raj R, Enright MP, Frangopol DM. System level partitioning approach for analyzing the origins of variability in life prediction of tungsten filaments for incandescent lamps. Submitted for publication w4x Hasselman DPH. Ceramurgia 1979;4:147. w5x Kim S. Material properties of ceramic cutting tools. Key Eng Mater 1994;96:33᎐80. w6x Wayne SF, Buljan ST. Microstructure and wear resistance of silicon nitride composites. In: Jahanmir S, editor. Friction and wear of ceramics. New York: Marcel Dekker, Inc, 1994: 231᎐261. w7x Evans AG, Marshall DB. Wear mechanisms in ceramics. In: Rigney DA, editor. Fundamentals of friction and wear of materials. Metals Park, OH: ASM, 1981:439᎐452.

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